// Numbas version: exam_results_page_options
{"name": "Integration by parts 1 with limits", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["s3", "c", "b", "m", "i1", "i0"], "name": "Integration by parts 1 with limits", "tags": ["algebraic manipulation", "Calculus", "calculus", "indefinite integration", "integrals", "integration", "integration by parts", "steps", "Steps"], "advice": "<p>The formula for integrating by parts is</p>\n<p>\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]</p>\n<p>We choose $u = x$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{({b}*x+{c})^{m}}$.</p>\n<p>So $\\displaystyle \\frac{du}{dx}$ = $1$ and $\\displaystyle v = \\simplify[std]{(1/{(m+1)*b})*({b}*x+{c})^{m+1}}$.</p>\n<p>Hence,<br/>\\[ \\begin{eqnarray*} \\displaystyle \\int^\\var{i1}_\\var{i0} \\simplify[std]{x*({b}x+{c})^{m}} dx &amp;=&amp; [uv]^\\var{i1}_\\var{i0} - \\int^\\var{i1}_\\var{i0} v \\frac{du}{dx} dx \\\\ &amp;=&amp; \\simplify[std]{[(x/{(m+1)*b})*({b}*x+{c})^{m+1}]}^\\var{i1}_\\var{i0}&nbsp;-\\simplify{ (1/{(m+1)*b})}*\\int^\\var{i1}_\\var{i0}(\\var{b}*x+\\var{c})^{\\var{m}+1} \\mathrm{d}x \\\\ &amp;=&amp; \\simplify[std]{[(x/{(m+1)*b})*({b}*x+{c})^{m+1}]}^\\var{i1}_\\var{i0} - \\simplify{(1/{(m+1)*(m+2)*b^2})*({b}*x+{c})^{m+2}}^\\var{i1}_\\var{i0} \\\\ &amp;=&amp;[\\simplify[std]{({b}*x+{c})^{m+1}/{(m+1)*(m+2)*b^2}*({b*(m+2)}x - ({b}x+{c}))}]^\\var{i1}_\\var{i0}\\\\ &amp;=&amp;[\\simplify[std]{({b}*x+{c})^{m+1}/{(m+1)*(m+2)*b^2}*({b*(m+1)}x - {c})}]^\\var{i1}_\\var{i0} \\end{eqnarray*}\\]<br/><br/></p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>The formula for integrating by parts is</p>\n<p>\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]<br/><br/></p>\n<p>What is the most suitable choice for $u$ and $\\frac{dv}{dx}$?</p>\n<p>$u=\\;$[[0]]</p>\n<p>$\\frac{dv}{dx}=\\;$[[1]]$+C$</p>\n<p></p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "x", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({b}x+{c})^{m}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Hence find $v =\\;$[[0]]&nbsp;</p>\n<p>and $\\frac{du}{dx} =\\;$[[1]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "(({b}x+{c})^({m}+1))/(({m}+1)*{b})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "1", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Hence $uv =\\;$[[0]]</p>\n<p>$\\int v\\frac{du}{dx}\\mathrm{d}x =\\;$[[1]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "x*(({b}x+{c})^({m}+1))/(({m}+1)*{b})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "(({b}x+{c})^({m}+2))/(({m}+1)*({m}+2)*{b}^2)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Hence $I = [uv]_\\var{i0}^\\var{i1}&nbsp;- \\int^\\var{i1}_\\var{i0} v\\frac{du}{dx}\\mathrm{d}x =\\;$[[1]]</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{b*(m+1)}*x-{c}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all,fractionnumbers", "scripts": {}, "answer": "({b}*{i1}+{c})^{m+1}/{(m+1)*(m+2)*b^2}*({b*(m+1)}*{i1} - {c})-({b}*{i0}+{c})^{m+1}/{(m+1)*(m+2)*b^2}*({b*(m+1)}*{i0} - {c})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Using integration by parts, find the following indefinite integral&nbsp;$I=\\displaystyle \\int^\\var{i1}_\\var{i0} \\simplify[std]{x*({b}x+{c})^{m}} dx $</p>\n<p>You need to know that for $n \\neq -1$:</p>\n<p>\\[ \\int (ax+b)^n dx = \\frac{1}{a(n+1)}(ax+b)^{n+1}+C\\]</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"c": {"definition": "s3*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "i0": {"definition": "floor(i1/2)", "templateType": "anything", "group": "Ungrouped variables", "name": "i0", "description": ""}, "m": {"definition": "random(2..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "i1": {"definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "i1", "description": ""}}, "metadata": {"notes": "\n    \t\t    \t\t<p><strong>3/08/2012:</strong></p>\n    \t\t    \t\t<p>Added tags.</p>\n    \t\t    \t\t<p>Added description.</p>\n    \t\t    \t\t<p>Checked calculation. OK.</p>\n    \t\t    \t\t<p>Got rid of redundant instructions about inputting constant of integration.</p>\n    \t\t    \t\t<p>Got rid of instruction re not inputting decimals - no restriction needed, so no forbidden strings.</p>\n    \t\t    \t\t<p>Penalised use of steps, 1 mark. Added message to that effect.</p>\n    \t\t    \t\t<p>Improved Advice display.</p>\n    \t\t    \n    \t\t", "description": "<p>Given that $\\displaystyle \\int^i_j x({ax+b)^{m}} dx=\\frac{1}{A}(ax+b)^{m+1}g(x)+C$ for a given integer $A$ and polynomial $g(x)$, find $g(x)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}